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Theorem csbima12 5191
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbima12  |-  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B )

Proof of Theorem csbima12
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ( F " B
) )
2 csbeq1 3352 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3352 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3imaeq12d 5175 . . . 4  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
51, 4eqeq12d 2486 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( F " B
)  =  ( [_ y  /  x ]_ F "
[_ y  /  x ]_ B )  <->  [_ A  /  x ]_ ( F " B )  =  (
[_ A  /  x ]_ F " [_ A  /  x ]_ B ) ) )
6 vex 3034 . . . 4  |-  y  e. 
_V
7 nfcsb1v 3365 . . . . 5  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3365 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
97, 8nfima 5182 . . . 4  |-  F/_ x
( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
10 csbeq1a 3358 . . . . 5  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3358 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11imaeq12d 5175 . . . 4  |-  ( x  =  y  ->  ( F " B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B ) )
136, 9, 12csbief 3374 . . 3  |-  [_ y  /  x ]_ ( F
" B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
145, 13vtoclg 3093 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
15 csbprc 3774 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F " B )  =  (/) )
16 csbprc 3774 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
1716imaeq2d 5174 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ F " [_ A  /  x ]_ B )  =  ( [_ A  /  x ]_ F " (/) ) )
18 ima0 5189 . . . 4  |-  ( [_ A  /  x ]_ F "
(/) )  =  (/)
1917, 18syl6req 2522 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
2015, 19eqtrd 2505 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F " B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
2114, 20pm2.61i 169 1  |-  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   [_csb 3349   (/)c0 3722   "cima 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852
This theorem is referenced by:  csbrn  5304  disjpreima  28271  csbpredg  31797  brtrclfv2  36390  sbcheg  36445
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